Rhode Island, the only state in New England that has not legalized gay marriage, began taking up the matter this week. The State House is expected to pass a bill Thursday that would allow anyone to marry “any eligible person regardless of gender.” But the measure faces resistance in the State Senate and its fate is uncertain. Teresa Paiva-Weed, a Democrat who is the Senate president, opposes same-sex marriage but has said she would allow a vote on it in committee. Supporters say that if it gets to the Senate floor, the measure will pass, but opponents are skeptical and state senators are being lobbied heavily by both sides. Supporters of same-sex marriage have sought to build on the momentum from last year’s elections, when voters in Maine, Maryland and Washington State approved it at the ballot box. Rhode Island is one of several states, including Delaware, Hawaii, Illinois, Minnesota and New Jersey, where supporters of gay marriage are trying to make legislative gains this year.